- Email: [email protected]
Heat transfer modelling in CO2 laser processing of optical fibres
1 July 1998
Optics Communications 152 Ž1998. 324–328
Heat transfer modelling in CO 2 laser processing of optical fibres A.J.C. Grellier, N.K. Zayer, C.N. Pannell
1
Applied Optics Group, School of Physical Sciences, UniÕersity of Kent at Canterbury, Canterbury, Kent CT2 7NR, UK Received 12 January 1998; revised 16 March 1998; accepted 17 March 1998
Abstract The diameter self-regulation of optical fibre tapers produced using a CO 2 laser has been examined theoretically and modelled numerically. Results of the simulation show that for small diameters, where the geometric model of absorption cannot be used, the Mie theory of absorption derived from the Maxwell equations exhibits structure due to resonances of the CO 2 laser radiation in the tapered fibre. At small diameters, the polarisation state of the CO 2 laser is important, and resonance effects cause the equilibrium temperaturerdiameter graph to exhibit oscillations. q 1998 Elsevier Science B.V. All rights reserved. PACS: 07.05.Tp; 42.55.Lt; 42.70.Ce; 42.81.Bm Keywords: CO 2 laser; Taper; Optical fibre; Silica; Absorption
1. Introduction All-fibre optical components have already been shown on numerous occasions to be more desirable than their bulk optic or integrated optic counterparts in fibre sensor systems due to Že.g.. lower insertion losses and requirement for adjustment. In-fibre acousto-optic devices based on tapered optical fibres and directional couplers have been shown to be potentially highly efficient due to the concentration of acoustic energy by the taper transition w1,2x. The fabrication of tapers requires special care. Apart the fact it has to take place in a clean environment, the taper shape has to be well-controlled to keep the adiabaticity w3x and the taper waist needs to be very uniform for optimum performance. Until now, most tapers are made using flames, however some of the flame parameters are not predictable and the
1
E-mail: [email protected]
turbulence of the flame can produce random perturbations in waist diameter. The use of a CO 2 laser can be in principle alleviate these problems. When a flame is used for tapering fibre, the process is stopped manually when the required diameter is reached. Using a CO 2 laser instead, the process can be ‘‘self-regulating’’. During the tapering process, when the waist elongates under the effect of heat and constant axial tension, the diameter decreases, and eventually reaches a point where insufficient energy is absorbed for the softening temperature of the optical fibre to be reached, hence elongation stops. Using this property and setting carefully the CO 2 beam and fibre parameters, the final diameter can be predicted before starting the fabrication process by modelling the heat transfers which occur during the tapering process. We also show that with tapers of the order of 55 mm or less the efficiency factor of absorption as predicted by vector diffraction theory ŽMie theory w4x. does not decrease monotonically with diameter, and can lead to ‘‘quantisation’’ of attainable diameters. This paper presents a model of the heat transfer within an optical fibre which incorporates the above effect and
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 1 6 4 - 3
A.J.C. Grellier et al.r Optics Communications 152 (1998) 324–328
also the temperature dependence of the various fibre parameters. It shows that the graph of equilibrium temperature of the fibre hot zone produced by the CO 2 laser against diameter is not monotonic, but exhibits structure due to resonances of the CO 2 laser radiation in the fibre and is polarisation dependent. Moreover the graph of the temperature against laser power shows that the steady temperature is proportional to the applied CO 2 laser power. We have two reasons for presenting this theoretical study. Firstly, it serves to highlight some interesting physics. Secondly, we feel it is of critical importance for any research group contemplating the purchase of a CO 2 laser for this type of application to have access to the type of information presented here.
Fig. 1. Schematic thermal picture of gain and loss in the fibre.
modelled using the approximation of a thin rod in the one-dimensional case. Validity using the 1-D case rather than the 3-D is verified by the thermal thinness condition w7x Žsee Table 1 for the explanation of the different terms.: d 2cr 4 Kt
2. Thermal equation Several papers w5,6x describe the usage of a CO 2 laser for tapering and point out some of its characteristics, however the previously derived thermal equation represented an insufficiently detailed model of the physical processes involved. The thermal equation representing a CO 2 laser with radiation incident perpendicular to the fibre axis has been
< 1.
This condition determines the time required for the interior of the fibre to reach a uniform temperature. In the present case, the temperature difference is less than 1% across the fibre for times greater than 50 ms for diameter less than 125 mm. Hence the thermal thinness condition being satisfied means that the fibre can effectively be considered uniform, or in other words, may be treated in the 1-D case as long as variation of the incident radiation is slow compared to 50 ms, which is the case here since the duration of the proposed tapering process is of the
Table 1 Nomenclature Žterms in SI units. Term
Definition
Value, units
A c d H
Cross-sectional area of the fibre Specific heat of the fibre w10x Diameter of the fibre Surface conductance or convection heat-transfer coefficient w6,8x Intensity of CO 2 laser Extinction coefficient w9x Thermal conductivity of the fibre w10x Refractive index of the silica w11x Perimeter of the fibre Total output laser power Rate of heat generation from the laser on the fibre per unit volume Efficiency factor of absorption w4x Radius of the fibre Time Temperature Beam waist in the x, y direction resp. Energy absorption rate w4x Emissivity CO 2 laser wavelength Density of the fibre Stefan-Boltzman constant
3 = 10y1 4 –7 = 10y8 m2 y887.26 y 0.538ŽT y 1570. q 56'T J kgy1 Ky1 Ž0.1–125. = 10y6 m 418.68 W my2 Ky1
I k K n p Ptotal qŽ x . Qabs r t T wx , w y Wabs ´ l r s
325
W my2 , see Eq. Ž3. 0.0182 q 10.1 = 10y5 ŽT y 273.15. 1.213 q 0.00089T q 1.78 = 10y9 ŽT q 311.2. 3 W my1 Ky1 2.0–2.2 6.3 = 10y7 –9.4 = 10y4 m 0–10 W W my3 , see Eq. Ž7. dimensionless, see Eq. Ž4. Ž0.05–62.5. = 10y6 m seconds, s Tair s 293 K approximately 500 = 10y6 m W, see Eq. Ž4. expwyŽT y 293.r500 q 1x q 0.06 10.6 = 10y6 m 2200 kg my3 5.67 = 10y8 W my3 Ky4
A.J.C. Grellier et al.r Optics Communications 152 (1998) 324–328
326
order of minutes. Special care has been taken with the fibre parameters and their variations as a function of the temperature in the range 300–2000 K at 10.6 mm w4,6,8– 11x. From Fig. 1, we obtain the following thermal transfer relation w7x. q < x q E˙g s qc q E˙s q q < xq d x q qr ,
Ž1.
where Žthe explanation of the following terms is given in Table 1., rate of conduction
½
q < x sy w KA Ž d T r d x . x x q < xq d x sy w KA Ž d T r d x . x x q d x
rate of energy storage
qc s Hp Ž T yTair . d x E˙s s r cA Ž d T r d t . d x
rate of heat generation
E˙g s Aq Ž x . d x
rate of radiation
qr s s´ p Ž T 4 yTair4 . d x
rate of convection
Developing each term, Eq. Ž1. becomes
E 2T Ex
2
4H s dK
Ž T y Tair . q
qŽ x.
rc ET K Et
4s´ q dK
Ž T 4 y Tair4 .
. Ž2. K The term q Ž x . needs a bit more consideration since from our knowledge no derivation of a realistic heat generation has been presented before. First the intensity of the beam y
assuming the elliptical TEM 00 mode falling on the fibre is derived and gives IŽ x, y. s
2 Ptotal wx w y p
exp y
ž
2 x2 wx2
2 y2 q
wy2
/
.
Ž3.
The efficiency factor of absorption Qabs Ž Qabs has been derived from the Mie theory, scattering theory of light based on the Maxwell equations since diameters of the order of the CO 2 wavelength or less are encountered. depends on the complex refractive index of the silica Ž n q ik . and the fibre diameter d w4x. Hence it is one of the most important parameters in Eq. Ž2. Žincluded in the q Ž x . term, see Eq. Ž7.. and it is defined as Wabs Qabs s , Ž4. GI where G is the fibre cross sectional area projected onto a plane perpendicular to the incident beam, I is the incident intensity and Wabs is the energy absorption rate, see Table 1. From Fig. 1, for a small slice of width d x in the x direction, the absorbed power is from Eq. Ž4., qr
d Wabs s Qabs d x
Hyr I Ž y . d y.
Ž5.
Moreover, from the definition of q Ž x . Žsee Table 1.: d Wabs qŽ x. s . Ž6. p r 2d x Finally using Eq. Ž3. to integrate I in Eq. Ž5. and putting
Fig. 2. Experimental arrangement for tapering process.
A.J.C. Grellier et al.r Optics Communications 152 (1998) 324–328
327
the resulting term d Wabs in Eq. Ž6., we obtain the rate of heat generation per unit volume in the fibre, qŽ x. s
Qabs'2 Ptotal erf dr'2 wy
Ž
wxp 3r2 r 2
. exp
2 x2
ž / y
wx2
.
Ž7.
3. Result of the simulation The derived Eq. Ž2. is a partial second order differential equation which has been solved numerically using the Gear-Nordsieck finite difference method and a variant of the programme given in Appendix C of Ref. w4x for the absorption term Qabs . As for a flame-operated taper pulling rig, the CO 2 beam would translate periodically along the fibre and heat up a region Žthe hot zone .. In our taper rig, the CO 2 laser beam is focused on the fibre by a mirror of focal length equal to 0.3 m, resulting in a beam diameter of 1 mm, then scanned along the fibre with a galvanometer mirror which determines the length of the hot zone Žsee Fig. 2.. Fig. 3 represents the equilibrium temperature calculated by solving Eq. Ž2. for both the parallel and perpendicular states of polarisation of the CO 2 laser using the Mie theory model and the geometric theory using the following relation Žwhere I0 is the incident laser power and the other terms are explained in Table 1.
ž
I s I0 exp y
4p kd
l
/
for different diameters Ž0.1–80 mm. with an incident laser power of 5 W. The geometric theory formula used for Fig. 3 is based on the ray Žgeometrical. theory of absorption in a cylinder. If the softening point of the optical fibre is approximately 1700 K, the final diameter will be approximately 23.9 mm for the perpendicular polarisation, 29.8 mm for the parallel polarisation and 35.3 mm using the geometrical model. This latter model cannot be used in reality with these small diameters Žsmaller than 55 mm. at this wave-
Fig. 3. Variation of the fibre temperature as a function of its diameter for k s 0.015 and a CO 2 laser power of 5 W.
Fig. 4. Variation of the fibre temperature as a function of its diameter for the perpendicular polarisation and k s 0.015.
length Ž10.6 mm. but gives a good overview to judge the limit of the utilisation of the geometrical model which is in this case about 5 l. Moreover the Mie theory model exhibits larger value of absorption than the geometrical one in the diameter range Ž10–45 mm. with structure due to resonances of the CO 2 radiation inside the fibre. The simulation has been carried out for the perpendicular state of polarisation of the CO 2 laser at different laser powers in the diameter range 0.1–10 mm, two figures have been realised from these data, Figs. 4 and 5. Fig. 4 shows clearly that the diameters at which oscillation peaks occur are independent of the laser power but are dependent from Fig. 3 on the polarisation state of the CO 2 laser. Finally, we can observe from Fig. 5 that the temperature of the fibre for given diameters increases linearly with the CO 2 laser power. This figure can be very helpful since once the final diameter has been decided, the laser power to apply can be directly found.
Fig. 5. Variation of the fibre temperature as a function of the applied CO 2 laser power for the perpendicular polarisation and k s 0.015.
328
A.J.C. Grellier et al.r Optics Communications 152 (1998) 324–328
4. Discussions and conclusions Fibre tapers and couplers can be made with a CO 2 laser avoiding problems such as contamination and random turbulence-induced non-uniformity. We have pointed out for the first time the effects which can arise when tapering down to small fibre diameters, in particular the non-monotonic dependence of equilibrium temperature on fibre diameter. This offers the possibility of quantising the attainable final diameters and increasing the strength of the self-regulating mechanism, which is driven by the local gradient of the temperaturerdiameter graph. Further considerations on the model of heat transfer in a silica fibre must be underlined. The softening point of pure silica is approximately 2000 K and is strongly affected by the presence of dopants. The variation and more precisely the value of the extinction coefficient k, which mainly determined the self-regulation diameter, is uncertain in the literature, and in addition there is a very large variation of k in the 9–11 mm range w11x. In general the variation of the fibre parameters has been extrapolated due to the lack of data in the range 1500–2000 K. The radiation term also requires care, it is usually assumed that the silica has an emissivity of 1, a good approximation at room temperature. The emissivity tends to decrease, however, as the temperature increases, and it becomes very 4. important when the term ŽT 4 y Tair becomes large. In addition, emissivity for small grey body objects such as our optical fibre depends on the size of the object w12x. In conclusion, to increase the accuracy of the thermal modelling still further, more accurate estimates of the
relevant fibre parameters and their dependence on temperature and wavelength are needed. Acknowledgements The authors would like to thank Dr. W.A.B. Evans for his enlightening discussion on the FORTRAN simulation progamme and Dr. T.A. Birks Žof the University of Bath. for drawing our attention to Ref. w12x. References w1x D.O. Culverhouse, T.A. Birks, S.G. Farwell, J. Ward, P.St.J. Russell, IEEE Photon. Lett. 8 Ž1996. 1636. w2x D.O. Culverhouse, R.I. Laming, S.G. Farwell, T.A. Birks, M.N. Zervas, IEEE Photon. Techn. Lett. 9 Ž1997. 455. w3x J.D. Love, W.M. Henry, W.J. Stewards, R.J. Black, S. Lacroix, F. Ghontier, IEEE Proceeding-J 138 Ž1991. 343. w4x G.F. Bohren, D.R. Huffman, Absorption and Scattering by Small Particles, Wiley, 1983. w5x H. Yokota, E. Sugai, Y. Kashima, Y. Sasaki, OFS-11, Intern. Conf. on Optical Sensors, May 1996, pp. 494–497. w6x D.R. Fairbanks, SPIE Fiber Optic Components and Reliability 1580 Ž1991. 188. w7x G.E. Myers, Analytical Methods in Conduction heat transfer, McGraw Hill, 1971. w8x U.C. Paek, A.L. Weaver, Appl. Optics 14 Ž1975. 294. w9x A.D. McLahlan, F.P. Meyer, Appl. Optics 26 Ž1987. 1728. w10x Y.S. Touloukian, C.Y. Ho, Thermophysical Properties of Matter: The TPRC Data Series, Perdue University, 1979. w11x E.D. Palik, Handbook of Optical Constants of Solids, Academic Press, 1985. w12x R. Gardon, J. Am. Ceramic Society 8 Ž1956. 278.
Optics Communications 152 Ž1998. 324–328
Heat transfer modelling in CO 2 laser processing of optical fibres A.J.C. Grellier, N.K. Zayer, C.N. Pannell
1
Applied Optics Group, School of Physical Sciences, UniÕersity of Kent at Canterbury, Canterbury, Kent CT2 7NR, UK Received 12 January 1998; revised 16 March 1998; accepted 17 March 1998
Abstract The diameter self-regulation of optical fibre tapers produced using a CO 2 laser has been examined theoretically and modelled numerically. Results of the simulation show that for small diameters, where the geometric model of absorption cannot be used, the Mie theory of absorption derived from the Maxwell equations exhibits structure due to resonances of the CO 2 laser radiation in the tapered fibre. At small diameters, the polarisation state of the CO 2 laser is important, and resonance effects cause the equilibrium temperaturerdiameter graph to exhibit oscillations. q 1998 Elsevier Science B.V. All rights reserved. PACS: 07.05.Tp; 42.55.Lt; 42.70.Ce; 42.81.Bm Keywords: CO 2 laser; Taper; Optical fibre; Silica; Absorption
1. Introduction All-fibre optical components have already been shown on numerous occasions to be more desirable than their bulk optic or integrated optic counterparts in fibre sensor systems due to Že.g.. lower insertion losses and requirement for adjustment. In-fibre acousto-optic devices based on tapered optical fibres and directional couplers have been shown to be potentially highly efficient due to the concentration of acoustic energy by the taper transition w1,2x. The fabrication of tapers requires special care. Apart the fact it has to take place in a clean environment, the taper shape has to be well-controlled to keep the adiabaticity w3x and the taper waist needs to be very uniform for optimum performance. Until now, most tapers are made using flames, however some of the flame parameters are not predictable and the
1
E-mail: [email protected]
turbulence of the flame can produce random perturbations in waist diameter. The use of a CO 2 laser can be in principle alleviate these problems. When a flame is used for tapering fibre, the process is stopped manually when the required diameter is reached. Using a CO 2 laser instead, the process can be ‘‘self-regulating’’. During the tapering process, when the waist elongates under the effect of heat and constant axial tension, the diameter decreases, and eventually reaches a point where insufficient energy is absorbed for the softening temperature of the optical fibre to be reached, hence elongation stops. Using this property and setting carefully the CO 2 beam and fibre parameters, the final diameter can be predicted before starting the fabrication process by modelling the heat transfers which occur during the tapering process. We also show that with tapers of the order of 55 mm or less the efficiency factor of absorption as predicted by vector diffraction theory ŽMie theory w4x. does not decrease monotonically with diameter, and can lead to ‘‘quantisation’’ of attainable diameters. This paper presents a model of the heat transfer within an optical fibre which incorporates the above effect and
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 1 6 4 - 3
A.J.C. Grellier et al.r Optics Communications 152 (1998) 324–328
also the temperature dependence of the various fibre parameters. It shows that the graph of equilibrium temperature of the fibre hot zone produced by the CO 2 laser against diameter is not monotonic, but exhibits structure due to resonances of the CO 2 laser radiation in the fibre and is polarisation dependent. Moreover the graph of the temperature against laser power shows that the steady temperature is proportional to the applied CO 2 laser power. We have two reasons for presenting this theoretical study. Firstly, it serves to highlight some interesting physics. Secondly, we feel it is of critical importance for any research group contemplating the purchase of a CO 2 laser for this type of application to have access to the type of information presented here.
Fig. 1. Schematic thermal picture of gain and loss in the fibre.
modelled using the approximation of a thin rod in the one-dimensional case. Validity using the 1-D case rather than the 3-D is verified by the thermal thinness condition w7x Žsee Table 1 for the explanation of the different terms.: d 2cr 4 Kt
2. Thermal equation Several papers w5,6x describe the usage of a CO 2 laser for tapering and point out some of its characteristics, however the previously derived thermal equation represented an insufficiently detailed model of the physical processes involved. The thermal equation representing a CO 2 laser with radiation incident perpendicular to the fibre axis has been
< 1.
This condition determines the time required for the interior of the fibre to reach a uniform temperature. In the present case, the temperature difference is less than 1% across the fibre for times greater than 50 ms for diameter less than 125 mm. Hence the thermal thinness condition being satisfied means that the fibre can effectively be considered uniform, or in other words, may be treated in the 1-D case as long as variation of the incident radiation is slow compared to 50 ms, which is the case here since the duration of the proposed tapering process is of the
Table 1 Nomenclature Žterms in SI units. Term
Definition
Value, units
A c d H
Cross-sectional area of the fibre Specific heat of the fibre w10x Diameter of the fibre Surface conductance or convection heat-transfer coefficient w6,8x Intensity of CO 2 laser Extinction coefficient w9x Thermal conductivity of the fibre w10x Refractive index of the silica w11x Perimeter of the fibre Total output laser power Rate of heat generation from the laser on the fibre per unit volume Efficiency factor of absorption w4x Radius of the fibre Time Temperature Beam waist in the x, y direction resp. Energy absorption rate w4x Emissivity CO 2 laser wavelength Density of the fibre Stefan-Boltzman constant
3 = 10y1 4 –7 = 10y8 m2 y887.26 y 0.538ŽT y 1570. q 56'T J kgy1 Ky1 Ž0.1–125. = 10y6 m 418.68 W my2 Ky1
I k K n p Ptotal qŽ x . Qabs r t T wx , w y Wabs ´ l r s
325
W my2 , see Eq. Ž3. 0.0182 q 10.1 = 10y5 ŽT y 273.15. 1.213 q 0.00089T q 1.78 = 10y9 ŽT q 311.2. 3 W my1 Ky1 2.0–2.2 6.3 = 10y7 –9.4 = 10y4 m 0–10 W W my3 , see Eq. Ž7. dimensionless, see Eq. Ž4. Ž0.05–62.5. = 10y6 m seconds, s Tair s 293 K approximately 500 = 10y6 m W, see Eq. Ž4. expwyŽT y 293.r500 q 1x q 0.06 10.6 = 10y6 m 2200 kg my3 5.67 = 10y8 W my3 Ky4
A.J.C. Grellier et al.r Optics Communications 152 (1998) 324–328
326
order of minutes. Special care has been taken with the fibre parameters and their variations as a function of the temperature in the range 300–2000 K at 10.6 mm w4,6,8– 11x. From Fig. 1, we obtain the following thermal transfer relation w7x. q < x q E˙g s qc q E˙s q q < xq d x q qr ,
Ž1.
where Žthe explanation of the following terms is given in Table 1., rate of conduction
½
q < x sy w KA Ž d T r d x . x x q < xq d x sy w KA Ž d T r d x . x x q d x
rate of energy storage
qc s Hp Ž T yTair . d x E˙s s r cA Ž d T r d t . d x
rate of heat generation
E˙g s Aq Ž x . d x
rate of radiation
qr s s´ p Ž T 4 yTair4 . d x
rate of convection
Developing each term, Eq. Ž1. becomes
E 2T Ex
2
4H s dK
Ž T y Tair . q
qŽ x.
rc ET K Et
4s´ q dK
Ž T 4 y Tair4 .
. Ž2. K The term q Ž x . needs a bit more consideration since from our knowledge no derivation of a realistic heat generation has been presented before. First the intensity of the beam y
assuming the elliptical TEM 00 mode falling on the fibre is derived and gives IŽ x, y. s
2 Ptotal wx w y p
exp y
ž
2 x2 wx2
2 y2 q
wy2
/
.
Ž3.
The efficiency factor of absorption Qabs Ž Qabs has been derived from the Mie theory, scattering theory of light based on the Maxwell equations since diameters of the order of the CO 2 wavelength or less are encountered. depends on the complex refractive index of the silica Ž n q ik . and the fibre diameter d w4x. Hence it is one of the most important parameters in Eq. Ž2. Žincluded in the q Ž x . term, see Eq. Ž7.. and it is defined as Wabs Qabs s , Ž4. GI where G is the fibre cross sectional area projected onto a plane perpendicular to the incident beam, I is the incident intensity and Wabs is the energy absorption rate, see Table 1. From Fig. 1, for a small slice of width d x in the x direction, the absorbed power is from Eq. Ž4., qr
d Wabs s Qabs d x
Hyr I Ž y . d y.
Ž5.
Moreover, from the definition of q Ž x . Žsee Table 1.: d Wabs qŽ x. s . Ž6. p r 2d x Finally using Eq. Ž3. to integrate I in Eq. Ž5. and putting
Fig. 2. Experimental arrangement for tapering process.
A.J.C. Grellier et al.r Optics Communications 152 (1998) 324–328
327
the resulting term d Wabs in Eq. Ž6., we obtain the rate of heat generation per unit volume in the fibre, qŽ x. s
Qabs'2 Ptotal erf dr'2 wy
Ž
wxp 3r2 r 2
. exp
2 x2
ž / y
wx2
.
Ž7.
3. Result of the simulation The derived Eq. Ž2. is a partial second order differential equation which has been solved numerically using the Gear-Nordsieck finite difference method and a variant of the programme given in Appendix C of Ref. w4x for the absorption term Qabs . As for a flame-operated taper pulling rig, the CO 2 beam would translate periodically along the fibre and heat up a region Žthe hot zone .. In our taper rig, the CO 2 laser beam is focused on the fibre by a mirror of focal length equal to 0.3 m, resulting in a beam diameter of 1 mm, then scanned along the fibre with a galvanometer mirror which determines the length of the hot zone Žsee Fig. 2.. Fig. 3 represents the equilibrium temperature calculated by solving Eq. Ž2. for both the parallel and perpendicular states of polarisation of the CO 2 laser using the Mie theory model and the geometric theory using the following relation Žwhere I0 is the incident laser power and the other terms are explained in Table 1.
ž
I s I0 exp y
4p kd
l
/
for different diameters Ž0.1–80 mm. with an incident laser power of 5 W. The geometric theory formula used for Fig. 3 is based on the ray Žgeometrical. theory of absorption in a cylinder. If the softening point of the optical fibre is approximately 1700 K, the final diameter will be approximately 23.9 mm for the perpendicular polarisation, 29.8 mm for the parallel polarisation and 35.3 mm using the geometrical model. This latter model cannot be used in reality with these small diameters Žsmaller than 55 mm. at this wave-
Fig. 3. Variation of the fibre temperature as a function of its diameter for k s 0.015 and a CO 2 laser power of 5 W.
Fig. 4. Variation of the fibre temperature as a function of its diameter for the perpendicular polarisation and k s 0.015.
length Ž10.6 mm. but gives a good overview to judge the limit of the utilisation of the geometrical model which is in this case about 5 l. Moreover the Mie theory model exhibits larger value of absorption than the geometrical one in the diameter range Ž10–45 mm. with structure due to resonances of the CO 2 radiation inside the fibre. The simulation has been carried out for the perpendicular state of polarisation of the CO 2 laser at different laser powers in the diameter range 0.1–10 mm, two figures have been realised from these data, Figs. 4 and 5. Fig. 4 shows clearly that the diameters at which oscillation peaks occur are independent of the laser power but are dependent from Fig. 3 on the polarisation state of the CO 2 laser. Finally, we can observe from Fig. 5 that the temperature of the fibre for given diameters increases linearly with the CO 2 laser power. This figure can be very helpful since once the final diameter has been decided, the laser power to apply can be directly found.
Fig. 5. Variation of the fibre temperature as a function of the applied CO 2 laser power for the perpendicular polarisation and k s 0.015.
328
A.J.C. Grellier et al.r Optics Communications 152 (1998) 324–328
4. Discussions and conclusions Fibre tapers and couplers can be made with a CO 2 laser avoiding problems such as contamination and random turbulence-induced non-uniformity. We have pointed out for the first time the effects which can arise when tapering down to small fibre diameters, in particular the non-monotonic dependence of equilibrium temperature on fibre diameter. This offers the possibility of quantising the attainable final diameters and increasing the strength of the self-regulating mechanism, which is driven by the local gradient of the temperaturerdiameter graph. Further considerations on the model of heat transfer in a silica fibre must be underlined. The softening point of pure silica is approximately 2000 K and is strongly affected by the presence of dopants. The variation and more precisely the value of the extinction coefficient k, which mainly determined the self-regulation diameter, is uncertain in the literature, and in addition there is a very large variation of k in the 9–11 mm range w11x. In general the variation of the fibre parameters has been extrapolated due to the lack of data in the range 1500–2000 K. The radiation term also requires care, it is usually assumed that the silica has an emissivity of 1, a good approximation at room temperature. The emissivity tends to decrease, however, as the temperature increases, and it becomes very 4. important when the term ŽT 4 y Tair becomes large. In addition, emissivity for small grey body objects such as our optical fibre depends on the size of the object w12x. In conclusion, to increase the accuracy of the thermal modelling still further, more accurate estimates of the
relevant fibre parameters and their dependence on temperature and wavelength are needed. Acknowledgements The authors would like to thank Dr. W.A.B. Evans for his enlightening discussion on the FORTRAN simulation progamme and Dr. T.A. Birks Žof the University of Bath. for drawing our attention to Ref. w12x. References w1x D.O. Culverhouse, T.A. Birks, S.G. Farwell, J. Ward, P.St.J. Russell, IEEE Photon. Lett. 8 Ž1996. 1636. w2x D.O. Culverhouse, R.I. Laming, S.G. Farwell, T.A. Birks, M.N. Zervas, IEEE Photon. Techn. Lett. 9 Ž1997. 455. w3x J.D. Love, W.M. Henry, W.J. Stewards, R.J. Black, S. Lacroix, F. Ghontier, IEEE Proceeding-J 138 Ž1991. 343. w4x G.F. Bohren, D.R. Huffman, Absorption and Scattering by Small Particles, Wiley, 1983. w5x H. Yokota, E. Sugai, Y. Kashima, Y. Sasaki, OFS-11, Intern. Conf. on Optical Sensors, May 1996, pp. 494–497. w6x D.R. Fairbanks, SPIE Fiber Optic Components and Reliability 1580 Ž1991. 188. w7x G.E. Myers, Analytical Methods in Conduction heat transfer, McGraw Hill, 1971. w8x U.C. Paek, A.L. Weaver, Appl. Optics 14 Ž1975. 294. w9x A.D. McLahlan, F.P. Meyer, Appl. Optics 26 Ž1987. 1728. w10x Y.S. Touloukian, C.Y. Ho, Thermophysical Properties of Matter: The TPRC Data Series, Perdue University, 1979. w11x E.D. Palik, Handbook of Optical Constants of Solids, Academic Press, 1985. w12x R. Gardon, J. Am. Ceramic Society 8 Ž1956. 278.